Question: $ F = \left[\begin{array}{rrr}1 & 2 & 2 \\ -2 & 0 & 2 \\ 1 & -2 & 4\end{array}\right]$ $ A = \left[\begin{array}{rrr}2 & -2 & -1 \\ 3 & -1 & 2 \\ 4 & 1 & -2\end{array}\right]$ Is $ F A$ defined?
Answer: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ F$ , have? How many rows does the second matrix, $ A$ , have? Since $ F$ has the same number of columns (3) as $ A$ has rows (3), $ F A$ is defined.